I am a number theorist interested in Galois representations and the Langlands program and related areas in representation theory and geometry.
I am an NSF postdoctoral research fellow at the University of Toronto with mentor Florian Herzig.
During the 2016-2017 academic year, I was a Member at the IAS.
I received my PhD from the University of Chicago working with Matthew Emerton.

My CV is available
pdf.

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Papers

*Preprint*. 2017. 7 pages.

[Abstract ±]
Let F/F^+ be a CM extension in which p is unramified and all places above p in F^+ split.
Let w be a place of F dividing p.
We show, under mild hypotheses related to the Taylor-Wiles method, that the m-torsion in the space of mod p automorphic
forms on a compact U(3) defined over F^+ with infinite level at w and appropriate level and coefficients at other places
is indecomposable when the associated Galois representation representation is irreducible and generic at w.

*Submitted*. 2017. 16 pages.

[Abstract ±]
Let F be a totally real field in which p is unramified.
We show, under a Taylor-Wiles hypothesis, that the m-torsion in the cohomology of Shimura curves with full congruence level at v, if nonzero, coincides with a representation constructed by Breuil-Paskunas.
In particular, it depends only on the restriction of the corresponding Galois representation to a decomposition group at v, and its Jordan-Holder factors appear with multiplicity one.
This builds on and extends work of the author with Morra and Schraen and independently of Hu-Wang, which proved these results when the Galois representation was additionally assumed to be tamely ramified.
The main new tool is a method for computing Taylor-Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti-Tate deformation rings and their intersection theory.

with

Bao V. Le Hung and

Brandon Levin
*Submitted.* 2016. 32 pages.

[Abstract ±]
We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of U(n) which are compact at infinity and split at places dividing p in generic situations. That is, we show that all modular weights for a mod p Galois representation are contained in the set predicted by Herzig. Under some additional hypotheses, we also show modularity of all the "obvious" weights.

with

Stefano Morra and

Benjamin Schraen
*Submitted.* 2016. 19 pages.

[Abstract ±]
Let F be a totally real field in which p is unramified. Let r be a modular
representation of the absolute Galois of F which satisfies the Taylor--Wiles
hypotheses and is tamely ramified and generic at a place v above p. Let m be
the corresponding Hecke eigensystem. We describe the m-torsion in the mod
p cohomology of Shimura curves with full congruence level at v as a GL_2(k_v)-representation.
In particular, it only depends on the restriction of r to the inertial subgroup at v,
and its Jordan--Holder
factors appear with multiplicity one. The main ingredients are a description
of the submodule structure of generic GL(2,q)-projective envelopes and the
multiplicity one results of [EGS15].

with

Bao V. Le Hung,

Brandon Levin, and

Stefano Morra
*Submitted.* 2016. 87 pages.

[Abstract ±]
We prove in generic situations that the lattice in a tame type induced by the completed
cohomology of a U(3)-arithmetic manifold is purely local, i.e., only depends on the Galois
representation at places above p. This is a generalization to GL3 of the lattice conjecture of Breuil.
In the process, we also prove the geometric Breuil-Mezard conjecture for (tamely) potentially crystalline
deformation rings with Hodge-Tate weights (0,1,2) as well as the Serre weight conjectures of
[Her09] over an unramified field extending the results of [LLHLM]. We also prove results in modular
representation theory about lattices in Deligne-Luzstig representations for the group GL(3,q).

with

Stefano Morra and

Chol Park
*Submitted.* 2016. 58 pages.

[Abstract ±]
Assume that r is a three dimensional modular mod p representation which is non-ordinary and nonsplit reducible (niveau 2) at a totally split place w above p.
Under mild hypotheses, we show that the isomorphism class of r locally at w is determined by the GL_3(Q_p)-action on the space of mod p algebraic
automorphic forms by using the refined Hecke action of [HLM]. We also give a nearly
optimal weight elimination result for niveau two Galois representations compatible with
the explicit conjectures of [Her09] and [GHS]. Moreover, we prove the modularity of
certain Serre weights, in particular, when the Fontaine-Laffaille invariant takes value infinity, our methods provide with the modularity of a certain shadow weight.

with

Florian Herzig and

Stefano Morra
Compositio Mathematica

**153** (2017), no. 11, 2215-2286.

[Abstract ±]
Suppose that r is an irreducible modular mod p Galois representation such that r is upper-triangular, maximally non-split, and generic locally at a totally split place w above p.
Under mild hypotheses, we show that r locally at w can be recovered from the GL_3(Q_p)-action on the space of mod p automorphic forms on a compact unitary group. On the way we prove results about weights in Serre's conjecture for r, show the existence of an ordinary lifting of r, and prove the freeness of certain Taylor-Wiles patched modules in this context. We also show the existence of many Galois representations r to which our main theorem applies.

with

Bao V. Le Hung,

Brandon Levin, and

Stefano Morra
*To appear in Inventiones.* 107 pages.

Journal
[Abstract ±]
We prove the weight part of Serre's conjecture in generic situations for forms of U(3) which are compact at infinity and split at places dividing p as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights (2,1,0) for K/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mezard conjectures hold for these deformation rings.

*To appear in Math. Annalen*. 30 pages.

[Abstract ±]
Under hypotheses required for the Taylor-Wiles method, we prove for forms of U(3) which are compact at
infinity that the lattice structure on upper alcove algebraic vectors or on principal series types given
by the lambda-isotypic part of completed cohomology is a local invariant of the Galois representation attached
to lambda when this Galois representation is residually irreducible locally at places dividing p. We combine
Hecke theory and weight cycling with the Taylor-Wiles method to establish crucial mod p multiplicity one
results for upper alcove algebraic vectors and principal series types.

with

Shelly Manber and

Shrenik Shah
International Journal of Number Theory

**6** (2010), no. 3, 625-653.

[Abstract ±]
We prove that logarithmic derivatives of certain twisted Hilbert class polynomials are holomorphic modular forms modulo p of
filtration p + 1. We derive p-adic information about twisted Hecke traces and Hilbert class polynomials. In this framework,
we formulate a precise criterion for p-divisibility of class numbers of imaginary quadratic fields in terms of the existence
of certain cusp forms modulo p. We explain the existence of infinite classes of congruent twisted Hecke traces with fixed
discriminant in terms of the factorization of the associated Hilbert class polynomial modulo p. Finally, we provide a new proof
of a theorem of Ogg classifying those p for which all supersingular j-invariants modulo p lie in F_p.